Simplify and expand the following expression: $ \dfrac{2t}{3t - 9}+\dfrac{t - 5}{t - 6} $
Answer: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(3t - 9)(t - 6)$ Multiply the first term by $\dfrac{t - 6}{t - 6}$ $ \begin{align*} \dfrac{2t}{3t - 9} \times \dfrac{t - 6}{t - 6} & = \dfrac{(2t)(t - 6)}{(3t - 9)(t - 6)} \\ & = \dfrac{2t^2 - 12t}{(3t - 9)(t - 6)}\end{align*} $ Multiply the second term by $\dfrac{3t - 9}{3t - 9}$ $ \begin{align*} \dfrac{t - 5}{t - 6} \times \dfrac{3t - 9}{3t - 9} & = \dfrac{(t - 5)(3t - 9)}{(t - 6)(3t - 9)} \\ & = \dfrac{3t^2 - 24t + 45}{(t - 6)(3t - 9)}\end{align*} $ Now we have: $ = \dfrac{2t^2 - 12t}{(3t - 9)(t - 6)} + \dfrac{3t^2 - 24t + 45}{(t - 6)(3t - 9)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{2t^2 - 12t + 3t^2 - 24t + 45}{(3t - 9)(t - 6)} $ $ = \dfrac{5t^2 - 36t + 45}{(3t - 9)(t - 6)}$ Expand the denominator: $ = \dfrac{5t^2 - 36t + 45}{3t^2 - 27t + 54}$